Bernoulli equation calculator
Trade pressure, speed and height between two points of a flowing fluid.
What is Bernoulli’s equation?
For steady flow of an incompressible fluid without friction, the sum p + ½ρv² + ρgh (pressure, kinetic energy per volume, potential energy per volume) is the same at every point along a streamline: p₁ + ½ρv₁² + ρgh₁ = p₂ + ½ρv₂² + ρgh₂ (OpenStax University Physics Vol. 1, Section 14.6, Equation 14.16). With the continuity equation A₁v₁ = A₂v₂ it tells you how pressure changes when a pipe narrows or rises: where the fluid speeds up at the same height, the pressure drops.
Example: a hose and a fire hose nozzle
OpenStax Example 14.6: water speeds up from 1.96 m/s in a hose to 25.5 m/s in the nozzle, where the pressure is atmospheric (1.01 × 10⁵ N/m²). Then p₁ = 1.01 × 10⁵ + ½ × 1000 × (25.5² − 1.96²) = 4.24 × 10⁵ N/m², about 424 kPa or 61.5 psi absolute. Example 14.7: a 6.40 cm fire hose carrying 40.0 L/s at 1.62 × 10⁶ N/m² gauge rises 10.0 m to a 3.00 cm nozzle. The speeds are 12.4 and 56.6 m/s and the nozzle pressure is about 0 gauge: −2.9 kPa with the book’s rounded speeds, −1.9 kPa unrounded. Switch to metric and use Pipe flow to try it.
What is Torricelli’s theorem?
Water leaving a hole a depth h below the surface of a large open tank has the speed v = √(2gh) — the same as an object that falls the distance h, and independent of the fluid’s density and of the size of the hole (OpenStax College Physics 2e, Section 12.3). A hole 5 ft down gives 17.9 ft/s (5.47 m/s). Real jets narrow just outside the hole, so the actual flow is lower than speed × hole area, which is the ideal figure the calculator shows.
Which fluid densities does it use?
Only OpenStax University Physics Vol. 1, Tables 14.1 and 14.2: fresh water 1,000 kg/m³ at 4 °C and 998.2 kg/m³ at 20 °C (Table 14.2), sea water 1,030 kg/m³ and olive oil 920 kg/m³ (Table 14.1, at 0 °C), and air 1.29 kg/m³ at 0 °C and 101.3 kPa. For anything else choose “Other” and type the density. Bernoulli’s equation assumes the density stays constant, which holds for liquids and for air only at speeds far below the speed of sound.
How is this different from the pipe velocity, head pressure and friction loss calculators?
The pipe velocity calculator only applies continuity: flow and diameter give speed. The head pressure calculator converts the height of a still water column to psi (ρgh alone). The friction loss calculator finds the pressure a real pipe run loses to friction. This one trades pressure, speed and height against each other in an ideal, frictionless flow; for a long pipe, subtract the friction loss from the pressure it gives. Pressures are gauge or absolute: pick one and use it for every entry.